Loading Posts & Category with Ajax

Sure, the get_template_part() function can be called multiple times referencing to different files, so if you have a file in the partials/-folder, all you have to do is add a second function call here.

function load_cat_posts () {
    $cat_id = absint($_REQUEST['cat']);
    $args = array (
       'cat' => $cat_id,
       'posts_per_page' => 10,
       'order' => 'DESC'
   );

    global $post;
    $posts = get_posts($args);

    ob_start ();

    foreach ( $posts as $post ) {
        setup_postdata( $post ); ?>

        <?php get_template_part( 'partials/listing', 'post'); ?>
        <?php get_template_part( 'partials/listing', 'category'); ?>

    <?php } wp_reset_postdata();

    $response = ob_get_contents();
    ob_end_clean();

    echo $response;
    die(
}

This will include the file partials/listing-category.php.

Edit: I misunderstood your question! To display the data related to the category itself, you can use the get_category function to fetch an object containing the following attributes:

stdClass Object
(
    [term_id] => 85
    [name] => Category Name
    [slug] => category-name
    [term_group] => 0
    [term_taxonomy_id] => 85
    [taxonomy] => category
    Loading Posts & Category with Ajax => 
    [parent] => 70
    [count] => 0
    [cat_ID] => 85
    [category_count] => 0
    [category_description] => 
    [cat_name] => Category Name
    [category_nicename] => category-name
    [category_parent] => 70
)

How you want to display this data is, of course, up to you.

function load_cat_posts () {
    $cat_id = absint($_REQUEST['cat']);
    $args = array (
       'cat' => $cat_id,
       'posts_per_page' => 10,
       'order' => 'DESC'
   );

   $category = get_category($cat_id);
   // ...
}